2021-2023年高考數(shù)學(xué)真題分類匯編專題04 導(dǎo)數(shù)及其應(yīng)用（解答題）（理）（原卷版）

專題04導(dǎo)數(shù)及其應(yīng)用（解答題）（理）知識(shí)點(diǎn)目錄知識(shí)點(diǎn)1：恒成立與有解問(wèn)題知識(shí)點(diǎn)2：極最值問(wèn)題知識(shí)點(diǎn)3：證明不等式知識(shí)點(diǎn)4：雙變量問(wèn)題（極值點(diǎn)偏移、拐點(diǎn)偏移）知識(shí)點(diǎn)5：零點(diǎn)問(wèn)題近三年高考真題知識(shí)點(diǎn)1：恒成立與有解問(wèn)題1．（2023?甲卷（理））已知SKIPIF1<0，SKIPIF1<0．（1）若SKIPIF1<0，討論SKIPIF1<0的單調(diào)性；（2）若SKIPIF1<0恒成立，求SKIPIF1<0的取值范圍．2．（2021?天津）已知SKIPIF1<0，函數(shù)SKIPIF1<0．（1）求曲線SKIPIF1<0在點(diǎn)SKIPIF1<0，SKIPIF1<0處的切線方程；（2）證明函數(shù)SKIPIF1<0存在唯一的極值點(diǎn)；（3）若SKIPIF1<0，使得SKIPIF1<0對(duì)任意的SKIPIF1<0恒成立，求實(shí)數(shù)SKIPIF1<0的取值范圍．3．（2023?上海）已知函數(shù)SKIPIF1<0，SKIPIF1<0（其中SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，若任意SKIPIF1<0，SKIPIF1<0均有SKIPIF1<0，則稱函數(shù)SKIPIF1<0是函數(shù)SKIPIF1<0的“控制函數(shù)”，且對(duì)所有滿足條件的函數(shù)SKIPIF1<0在SKIPIF1<0處取得的最小值記為SKIPIF1<0．（1）若SKIPIF1<0，SKIPIF1<0，試判斷函數(shù)SKIPIF1<0是否為函數(shù)SKIPIF1<0的“控制函數(shù)”，并說(shuō)明理由；（2）若SKIPIF1<0，曲線SKIPIF1<0在SKIPIF1<0處的切線為直線SKIPIF1<0，證明：函數(shù)SKIPIF1<0為函數(shù)SKIPIF1<0的“控制函數(shù)”，并求SKIPIF1<0的值；（3）若曲線SKIPIF1<0在SKIPIF1<0，SKIPIF1<0處的切線過(guò)點(diǎn)SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，證明：當(dāng)且僅當(dāng)SKIPIF1<0或SKIPIF1<0時(shí)，SKIPIF1<0（c）SKIPIF1<0（c）．知識(shí)點(diǎn)2：極最值問(wèn)題4．（2023·北京·統(tǒng)考高考真題）設(shè)函數(shù)SKIPIF1<0，曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程為SKIPIF1<0．(1)求SKIPIF1<0的值；(2)設(shè)函數(shù)SKIPIF1<0，求SKIPIF1<0的單調(diào)區(qū)間；(3)求SKIPIF1<0的極值點(diǎn)個(gè)數(shù)．5．（2023?新高考Ⅱ）（1）證明：當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0；參考答案（2）已知函數(shù)SKIPIF1<0，若SKIPIF1<0為SKIPIF1<0的極大值點(diǎn)，求SKIPIF1<0的取值范圍．6．（2023?乙卷（理））已知函數(shù)SKIPIF1<0．（1）當(dāng)SKIPIF1<0時(shí)，求曲線SKIPIF1<0在點(diǎn)SKIPIF1<0，SKIPIF1<0（1）SKIPIF1<0處的切線方程；（2）是否存在SKIPIF1<0，SKIPIF1<0，使得曲線SKIPIF1<0關(guān)于直線SKIPIF1<0對(duì)稱，若存在，求SKIPIF1<0，SKIPIF1<0的值，若不存在，說(shuō)明理由；（3）若SKIPIF1<0在SKIPIF1<0存在極值，求SKIPIF1<0的取值范圍．知識(shí)點(diǎn)3：證明不等式7．（2022?新高考Ⅱ）已知函數(shù)SKIPIF1<0．（1）當(dāng)SKIPIF1<0時(shí)，討論SKIPIF1<0的單調(diào)性；（2）當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，求SKIPIF1<0的取值范圍；（3）設(shè)SKIPIF1<0，證明：SKIPIF1<0．8．（2023?新高考Ⅰ）已知函數(shù)SKIPIF1<0．（1）討論SKIPIF1<0的單調(diào)性；（2）證明：當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0．9．（2021?乙卷（理））已知函數(shù)SKIPIF1<0，已知SKIPIF1<0是函數(shù)SKIPIF1<0SKIPIF1<0的極值點(diǎn)．（1）求SKIPIF1<0；（2）設(shè)函數(shù)SKIPIF1<0．證明：SKIPIF1<0．10．（2023?天津）已知函數(shù)SKIPIF1<0．（Ⅰ）求曲線SKIPIF1<0在SKIPIF1<0處的切線斜率；（Ⅱ）當(dāng)SKIPIF1<0時(shí)，求證：SKIPIF1<0；（Ⅲ）證明：SKIPIF1<0．知識(shí)點(diǎn)4：雙變量問(wèn)題（極值點(diǎn)偏移、拐點(diǎn)偏移）11．（2021?新高考Ⅰ）已知函數(shù)SKIPIF1<0．（1）討論SKIPIF1<0的單調(diào)性；（2）設(shè)SKIPIF1<0，SKIPIF1<0為兩個(gè)不相等的正數(shù)，且SKIPIF1<0，證明：SKIPIF1<0．12．（2022?天津）已知SKIPIF1<0，SKIPIF1<0，函數(shù)SKIPIF1<0，SKIPIF1<0．（1）求函數(shù)SKIPIF1<0在SKIPIF1<0，SKIPIF1<0處的切線方程；（2）若SKIPIF1<0和SKIPIF1<0有公共點(diǎn)．（?。┊?dāng)SKIPIF1<0時(shí)，求SKIPIF1<0的取值范圍；（ⅱ）求證：SKIPIF1<0．13．（2022?浙江）設(shè)函數(shù)SKIPIF1<0．（Ⅰ）求SKIPIF1<0的單調(diào)區(qū)間；（Ⅱ）已知SKIPIF1<0，SKIPIF1<0，曲線SKIPIF1<0上不同的三點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0處的切線都經(jīng)過(guò)點(diǎn)SKIPIF1<0．證明：（?。┤鬝KIPIF1<0，則SKIPIF1<0（a）SKIPIF1<0；（ⅱ）若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0．（注SKIPIF1<0是自然對(duì)數(shù)的底數(shù)）14．（2022?北京）已知函數(shù)SKIPIF1<0．（Ⅰ）求曲線SKIPIF1<0在點(diǎn)SKIPIF1<0，SKIPIF1<0處的切線方程；（Ⅱ）設(shè)SKIPIF1<0，討論函數(shù)SKIPIF1<0在SKIPIF1<0，SKIPIF1<0上的單調(diào)性；（Ⅲ）證明：對(duì)任意的SKIPIF1<0，SKIPIF1<0，有SKIPIF1<0．知識(shí)點(diǎn)5：零點(diǎn)問(wèn)題15．（2022?甲卷（理））已知函數(shù)SKIPIF1<0．（1）若SKIPIF1<0，求SKIPIF1<0的取值范圍；（2）證明：若SKIPIF1<0有兩個(gè)零點(diǎn)SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0．16．（2022?新高考Ⅰ）已知函數(shù)SKIPIF1<0和SKIPIF1<0有相同的最小值．（1）求SKIPIF1<0；（2）證明：存在直線SKIPIF1<0，其與兩條曲線SKIPIF1<0和SKIPIF1<0共有三個(gè)不同的交點(diǎn)，并且從左到右的三個(gè)交點(diǎn)的橫坐標(biāo)成等差數(shù)列．17．（2021?新高考Ⅱ）已知函數(shù)SKIPIF1<0．（Ⅰ）討論SKIPIF1<0的單調(diào)性；（Ⅱ）從下面兩個(gè)條件中選一個(gè)，證明：SKIPIF1<0恰有一個(gè)零點(diǎn)．①SKIPIF1<0，SKIPIF1<0；②SKIPIF1<0，SKIPIF1<0．18．（2021?浙江）設(shè)SKIPIF1<0，SKIPIF1<0為實(shí)數(shù)，且SKIPIF1<0，函數(shù)SKIPIF1<0．（Ⅰ）求函數(shù)SKIPIF1<0的單調(diào)區(qū)間；（Ⅱ）若對(duì)任意SKIPIF1<0，函數(shù)SKIPIF1<0有兩個(gè)不同的零點(diǎn)，求SKIPIF1<0的取值范圍；（Ⅲ）當(dāng)SKIPIF1<0時(shí)，證明：對(duì)任意SKIPIF1<0，函數(shù)SKIPIF1<0有兩個(gè)不同的零點(diǎn)SKIPIF1<0，SKIPIF1<0，滿足SKIPIF1<0．（注SKIPIF1<0是自然對(duì)數(shù)的底數(shù)）19．（2021?甲卷（理））已知SKIPIF1<0